Joakim Petersson

Numerical instabilities in topology optimization : A survey on procedures dealing with checkerboards, mesh-dependencies and local minima

In this paper we seek to summarize the current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method. The checkerboard problem refers to the formation of regions of alternating solid and void elements ordered in a checkerboard-like fashion. The mesh-dependence problem refers to obtaining qualitatively different solutions for different mesh-sizes or discretizations. Local minima refers to the problem of obtaining different solutions to the same discretized problem when choosing different algorithmic parameters. We review the current knowledge on why and when these problems appear, and we list the methods with which they can be avoided and discuss their advantages and disadvantages.

Topology optimization using regularized intermediate density control

We consider topology optimization of elastic continua. The elasticity tensor is assumed to depend linearly on the design function (density) as in the variable thickness sheet problem. In order to get “black–white” design pictures, the intermediate density values are controlled by an explicit constraint. This constraint is regularized by including a compact and linear operator S to guarantee existence of solutions. A proof of convergence of the finite element (FE) discretized optimization problems solutions to exact ones is also given, so the method is not prone to numerical anomalies such as mesh dependence or checkerboards. The procedure is illustrated in some minimum compliance examples where S is chosen to be a classical convolution-type operator. The FE-discretized optimization problems are solved by sequential convex approximations.

Some Convergence Results in Perimeter-Controlled Topology Optimization

Abstract The computation of optimal topologies of elastic continuum structures using a constraint on the ‘perimeter’ is investigated. Predicting macroscopic ‘black-white’ topologies without the use of homogenization techniques, this approach is presently one of the most attractive approaches in topology optimization. Mathematical justifications are given for both the relaxation of the discrete-value constraint on the design variable and for the finite element discretizations. It turns out that the way in which the perimeter has been calculated to date, the numerical results will not approximate the intended original problem, but one with a ‘taxi-cab’ perimeter which measures lengths of structural edges after projection onto the coordinate axes.

Large-scale topology optimization in 3D using parallel computing

We consider large-scale topology optimization of elastic continua in 3D using the regularized intermediate density control introduced in [1]. The nested approach is used, i.e., equilibrium is solved at each iteration. To get a high-quality resolution of realistic designs in 3D, problems involving several hundreds of thousand finite elements are solved. In order to deal with problems of this size, parallel computing is used in combination with domain decomposition. The equilibrium equations are solved by a preconditioned conjugate gradient algorithm and the optimization part is solved using sequential convex programming. Several numerical results obtained from computations performed on a Cray parallel computer are presented.

Optimization of Structures in Unilateral Contact

This article gives a review of optimization of structures in mechanical contact. Emphasis is put on linear elastic structures in frictionless contact. In particular, for optimization problems where an energy objective is used, a unified framework is given in parallel with the review. Papers related to optimal control of variational inequalities or dealing with pure sensitivity analysis are treated in less detail. Problems involving friction are also reviewed at a less detailed level. It is explained why structural optimization problems involving contact cannot be treated within classical smooth optimization theory and how they relate to modern fields such as nonsmooth optimization and mathematical programs with equilibrium constraints (MPECs). Throughout the article, discrete and continuous problems are treated in parallel. This review article includes 106 references.

Topology optimization of flow networks

The field of topology optimization is well developed for load carrying trusses, but so far not for other similar network problems. The present paper is a first study in the direction of topology op ...

A Finite Element Analysis of Optimal Variable Thickness Sheets

A quasi-mixed finite element (FE) method for maximum stiffness of variable thickness sheets is analyzed. The displacement is approximated with nine node Lagrange quadrilateral elements, and the thickness is approximated as elementwise constant. One is guaranteed that the FE displacement solutions will converge in

On continuity of the design-to-state mappings for trusses with variable topology

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TOPOLOGY OPTIMIZATION OF SHEETS IN CONTACT BY A SUBGRADIENT METHOD

, but in an example it is shown that, in general, one cannot expect any subsequence of the FE thickness solutions to converge in any

Existence and Continuity of Optimal Solutions to some Structural Topology Optimization Problems Including Unilateral Constraints and Stochastic Loads

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